Applications of Matrices and Determinants

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Step 1: For a 2×2 matrix [[a, b], [c, d]], adjoint = [[d, -b], [-c, a]] Step 2: Replace a=2, b=-1, c=3, d=4 Step 3: adj(A) = [[4, 1], [-3, 2]] Answer: adj(A) = [[4, 1], [-3, 2]] Note: Adjoint is the …

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Step 1: Calculate determinant |A| = 2(4) - (-1)(3) = 8 + 3 = 11 Step 2: Find adj(A) = [[4, 1], [-3, 2]] (from previous problem) Step 3: Apply formula A⁻¹ = (1/11) × [[4, 1], [-3, 2]] Step 4: A⁻¹ = [[4…

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Step 1: Write in matrix form AX = B where A = [[2, 3], [1, 2]], X = [[x], [y]], B = [[8], [5]] Step 2: Find |A| = 2(2) - 3(1) = 4 - 3 = 1 ≠ 0, so A⁻¹ exists Step 3: Find adj(A) = [[2, -3], [-1, 2]] St…

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Step 1: Identify Δ = determinant of coefficient matrix = |3, 2; 1, -1| = -3 - 2 = -5 Step 2: Calculate Δ₁ (replace x-column with constants) = |7, 2; 2, -1| = -7 - 4 = -11 Step 3: Calculate Δ₂ (replace…

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Use Cramer's Rule when: 1. The number of equations equals the number of unknowns (square system) 2. The determinant of the coefficient matrix Δ ≠ 0 (non-singular matrix) 3. You need a unique solution …

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Step 1: First matrix is [[1, 2, 3, 13], [2, -1, 1, 5], [3, 1, -1, 4]] Step 2: R₂ → R₂ - 2R₁: [[1, 2, 3, 13], [0, -5, -5, -21], [3, 1, -1, 4]] Step 3: R₃ → R₃ - 3R₁: [[1, 2, 3, 13], [0, -5, -5, -21], […

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Step 1: Reduce to row-echelon form: [[1, 2, 3, 13], [0, -5, -5, -21], [0, 0, -5, -14]] Step 2: From row 3: -5z = -14 → z = 14/5 Step 3: Substitute z into row 2: -5y - 5(14/5) = -21 → -5y - 14 = -21 → …

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Step 1: Original matrix A = [[1, 2, 3], [2, 4, 6], [3, 6, 9]] Step 2: R₂ → R₂ - 2R₁: [[1, 2, 3], [0, 0, 0], [3, 6, 9]] Step 3: R₃ → R₃ - 3R₁: [[1, 2, 3], [0, 0, 0], [0, 0, 0]] Step 4: Row-echelon form…